Recoverability conditions can be found by ignoring the low-rank nature of all the \(\mathbf {Z}_\text{i}\)  and focusing on the fact that the concatenation of the vectorized forms of all these matrices \(\mathbf {z}= \mathrm {vec}([\mathbf {Z}_1, \ldots , \mathbf {Z}_\text{R}])\)  is effectively a sparse vector. In this way, one can think of recovering a sparse vector \(\mathbf {z}\)  satisfying \(\mathbf {y}= [\mathbf {M}_1, \ldots , \mathbf {M}_\text{R}] \, \mathbf {z}\)  and derive recoverability conditions in this basic setting [1]}. A more sophisticated analysis accounting for the structure of the linear operators \(\mathbf {M}_\text{i}\)  and the low-rank nature of \(\mathbf {Z}_\text{i}\)  is our main theoretical contribution, which is presented in S:Recovery.
